Actuarial Statistics With Generalized Linear Mixed

Models

Katrien Antonio∗† Jan Beirlant ‡

Revision Feburary 2006

Abstract

Over the last decade the use of generalized linear models (GLMs) in actuarial statis-tics received a lot of attention, starting from the actuarial illustrations in the stan-dard text by McCullagh & Nelder (1989). Traditional GLMs however model asample of independent random variables. Since actuaries very often have repeatedmeasurements or longitudinal data (i.e. repeated measurements over time) at theirdisposal, this article considers statistical techniques to model such data within theframework of GLMs. Use is made of generalized linear mixed models (GLMMs)which model a transformation of the mean as a linear function of both fixed andrandom effects. The likelihood and Bayesian approaches to GLMMs are explained.The models are illustrated by considering classical credibility models and more gen-eral regression models for non-life ratemaking in the context of GLMMs. Details oncomputation and implementation (in SAS and WinBugs) are provided.

Keywords: non-life ratemaking, credibility, Bayesian statistics, longitudinal data,generalized linear mixed models.

∗Corresponding author: Katrien.Antonio@econ.kuleuven.be (phone: +32 (0) 16 32 67 69)†Ph. D. student, University Center for Statistics, W. de Croylaan 54, 3001 Heverlee, Belgium.‡University Center for Statistics, KU Leuven, W. de Croylaan 54, 3001 Heverlee, Belgium.

1

1 Introduction

Over the last decade generalized linear models (GLMs) became a common statistical tool

to model actuarial data. Starting from the actuarial illustrations in the standard text by

McCullagh & Nelder (1989), over applications of GLMs in loss reserving, credibility and

mortality forecasting, a whole scala of actuarial problems can be enumerated where these

models are useful (see Haberman & Renshaw, 1996, for an overview). The main merits of

GLMs are twofold. Firstly, regression is no longer restricted to normal data, but extended

to distributions from the exponential family. This enables appropriate modelling of, for

instance, frequency counts, skewed or binary data. Secondly, a GLM models the additive

effect of explanatory variables on a transformation of the mean, instead of the mean itself.

Standard GLMs require a sample of independent random variables. In many actuarial

and general statistical problems however the assumption of independence is not fulfilled.

Longitudinal, spatial or (more general) clustered data are examples of data structures

where this assumption is doubtful. This paper puts focus on repeated measurements

and, more specific, longitudinal data, which are repeated measurements on a group of

‘subjects’ over time. The interpretation of ‘subject’ depends on the context; in our illus-

trations policyholders and groups of policyholders (risk classes) are considered. Since they

share subject-specific characteristics, observations on the same subject over time often are

substantively correlated and require an appropriate toolbox for statistical modelling.

Two popular extensions of GLMs for correlated data are the so-called marginal models

based on generalized estimating equations (GEEs) on the one hand and the generalized

linear mixed models (GLMMs) on the other hand. Marginal models are only mentioned

indirectly and do not constitute the main topic of this paper. We focus on the character-

istics and applications of GLMMs.

Since the appearance of Laird & Ware (1982) linear mixed models are widely used

(e.g. in bio- and environmental statistics) to model longitudinal data. Mixed models

extend classical linear regression models by including random or subject-specific effects –

next to the (traditional) fixed effects – in the structure for the mean. For distributions

from the exponential family, GLMMs extend GLMs by including random effects in the

linear predictor. The random effects not only determine the correlation structure between

observations on the same subject, they also take account of heterogeneity among subjects,

due to unobserved characteristics.

In an actuarial context Frees et al. (1999, 2001) provide an excellent introduction to

linear mixed models and their applications in ratemaking. We will revisit some of their

illustrations in the framework of generalized linear mixed models. Using likelihood-based

hierarchical generalized linear models, Nelder & Verrall (1997) give an interpretation of

traditional credibility models in the framework of GLMs. Hierarchical generalized linear

models are GLMMs with random effects that are not necessarily normally distributed; an

1

assumption that is traditionally made. Since the statistical expertise concerning GLMMs

is more extensive, this paper puts focus on these models. Apart from traditional credibility

models, various other applications are considered as well.

Because both are valuable, estimation and inference in a likelihood-based as well as a

Bayesian framework is discussed. In a commercial software package like SAS 1, the results

of a likelihood-based analysis are easy to obtain with standard statistical procedures. Our

Bayesian implementation relies on Markov Chain Monte Carlo (MCMC) simulations. The

results of the likelihood-based analysis can be used for instance to choose starting values

for the chains and to check the reasonableness of the results. In an actuarial context, an

important advantage of the Bayesian approach is that it yields the posterior predictive

distribution of quantities of interest.

Spatial data and generalized additive mixed models (GAMMs) are outside the scope

of this paper. Recent work by Denuit & Lang (2004) and Fahrmeir et al. (2003) considers

a Bayesian implementation of a generalized additive model (GAM) for insurance data

with a spatial structure.

The paper is organized as follows. Section 2 introduces two motivating data sets which

will be analyzed later on. In Section 3 we first recall (briefly) the basic concepts of GLMs

and linear mixed models. Afterwards GLMMs are introduced and both maximum likeli-

hood (i.e. pseudo-likelihood or penalized quasi-likelihood and (adaptive) Gauss-Hermite

quadrature) and Bayesian estimation are discussed. In Section 4 we start with the formu-

lation of basic credibility models as particular GLMMs. The crossed classification model

of Dannenburg et al. (1996) is illustrated on a data set. Afterwards, illustrations on

workers’ compensation insurance data are fully explained. Other interesting applications

of GLMMs, for instance in credit risk modelling, are briefly sketched. Finally, Section 5

concludes.

2 Motivating actuarial examples

Two data sets from workers’ compensation insurance are considered. With the intro-

duction of these data we want to motivate the need for an extension of GLMs that is

appropriate to model correlated –here: longitudinal– data.

2.1 Workers’ Compensation Insurance: Frequencies

The data are taken from Klugman (1992). Here 133 occupation or risk classes are fol-

lowed over a period of 7 years. Frequency counts in workers’ compensation insurance are

observed on a yearly basis. Let Count denote the response variable of interest. Possible

explanatory variables are Year and Payroll, a measure of exposure denoting scaled payroll

1SAS is a commercial software package, see http://www.sas.com.

2

totals adjusted for inflation. Klugman (1992) and later on also Scollnik (1996) and Makov

et al. (1996) have analyzed these data in a Bayesian context (with no explicit formulation

as a GLMM). Exploratory plots for the raw data (not adjusted for exposure) are given

in Figure 1 and 2. A histogram of the complete data set and boxplots of the yearly data

are shown in Figure 1. The right panel in Figure 2 plots selected response profiles over

time and indicates the heterogeneity across the risk classes in the data set. Assuming in-

dependence between observations, a Poisson regression model would be a suitable choice

since the data are counts. However, the left panel in Figure 2 clearly shows substantive

correlation between subsequent observations on the same risk class. Our analysis uses a

Poisson GLMM, a choice that will be motivated further in Section 4.

0 50 100 150 200

020

040

060

0

Workers’ Compensation Data (Frequencies)

Count

050

100

150

200

1 2 3 4 5 6 7

Workers’ Compensation Data (Frequencies)

Year

Cou

nt

Figure 1: Histogram of Count (whole data set) and boxplots of Count for the 7 years in

the study, workers’ compensation data (frequencies).

Year1

0 50 100 150 0 50 100 150 200 0 50 100 150

050

150

050

100

150

Year2

Year3

050

100

050

150

Year4

Year5

050

150

050

150

Year6

0 50 100 200 0 50 100 150 0 50 100 200 0 50 100 150

050

150

Year7

Workers’ Compensation Data (Frequencies)

Year

Cou

nt

050

100

150

200

1 2 3 4 5 6 7

Figure 2: Scatterplot matrix of frequency counts (Year i against Year j for i, j = 1, . . . , 7)

(left) and selection of response profiles (right), workers’ compensation data (frequencies).

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2.2 Workers’ Compensation Insurance: Losses

We next consider a data set from the National Council on Compensation Insurance (USA)

containing losses due to permanent partial disability. As with the previous illustration,

Klugman (1992) already analyzed these data to illustrate the possibilities of Bayesian

statistics in non-life ratemaking. 121 occupation or risk classes are observed over a period

of 7 years. The variable Loss gives the amount of money paid out (on a yearly basis).

Possible explanatory variables are Year and Payroll. The data also appeared in Frees

et al. (2001) where the pure premium, PP=Loss/Payroll, was used as response variable.

Figure 3 contains exploratory plots for the variable Loss. The right-skewness of boxplot

and histogram is apparent. Figure 4 reveals substantive correlation between subsequent

observations on the same risk class. In Section 4 a gamma GLMM is used to model these

longitudinal data in the GLM framework.

0 5*10^6 10^7 1.5*10^7 2*10^7

010

020

030

040

050

060

0

Workers’ Compensation Data (Losses)

LOSS

05*

10^6

10^7

1.5*

10^7

2*10

^7

1 2 3 4 5 6 7

Workers’ Compensation Data (Losses)

Year

LOS

S

Figure 3: Histogram of Loss (whole database) and boxplots of Loss for the 7 years in the

study, workers’ compensation data (losses).

3 GLMMs for actuarial data

3.1 Basic concepts of GLMs and linear mixed models

For the reader’s convenience, we provide a short summary of the main characteristics of

GLMs on the one hand and mixed models on the other hand, before introducing their

extension to GLMMs. For a broad introduction to generalized linear models (GLMs) we

refer to McCullagh & Nelder (1989). Numerous illustrations of the use of GLMs in typi-

cal problems from actuarial statistics are available; see e.g. Haberman & Renshaw (1996).

Full details on linear mixed models can be found in Verbeke & Molenberghs (2000) or

Demidenko (2004). Frees et al. (1999) give a good introduction to the possibilities of

4

Year1

0 4*10^6 10^7 0 5*10^6 1.5*10^7 0 10^7

010

^7

06*

10^6

Year2

Year3

010

^7

010

^7

Year4

Year5

010

^7

010

^7

Year6

0 5*10^6 1.5*10^7 0 5*10^6 1.5*10^7 0 10^7 2*10^7 0 4*10^6 10^7

06*

10^6

Year7

Workers’ Compensation Data (Losses)

Year

Loss

05*

10^6

10^7

1.5*

10^7

2*10

^7

1 2 3 4 5 6 7

Figure 4: Scatterplot matrix of losses, Year i against Year j for i, j = 1, . . . , 7 (left) and

selection of response profiles (right), workers’ compensation data (losses).

linear mixed models in actuarial statistics.

Generalized Linear Models (GLMs)

A first important feature of GLMs is that they extend the framework of general (nor-

mal) linear models to the class of distributions from the exponential family. A whole

variety of possible outcome measures (like counts, binary and skewed data) can be mod-

elled within this framework. This paper uses the canonical form specification of densities

from the exponential family, namely

f(y) = exp

(yθ − ψ(θ)

φ+ c(y, φ)

)(1)

where ψ(.) and c(.) are known functions, θ is the natural and φ the scale parameter.

Let Y1, . . . , Yn be independent random variables with a distribution from the exponential

family. The following well-known relations can easily be proven for such distributions

µi = E[Yi] = ψ′(θi) and Var[Yi] = φψ

′′(θi) = φV (µi) (2)

where derivatives are with respect to θ and V (.) is called the variance function. This

function captures the relationship, if any exists, between the mean and variance of Y .

A second important feature of GLMs is that they provide a way around the transfor-

mation of data. Instead of a transformed data vector, a transformation of the mean is

modelled as a linear function of explanatory variables. In this way

g(µi) = ηi = (Xβ)i (3)

where β = (β1, . . . , βp)′

contains the model parameters and X (n × p) is the design

matrix. g is the link function and ηi is the ith element of the so-called linear predictor.

5

The unknown but fixed regression parameters in β are estimated by solving the maximum

likelihood equations with an iterative numerical technique (such as Newton-Raphson).

Likelihood ratio and Wald tests are used for inference purposes. If the scale parameter φ

is unknown, it can be estimated either by maximum likelihood or by dividing the deviance

or Pearson’s chi-square statistic by its degrees of freedom.

GLMs are well suited to model various outcomes from cross-sectional experiments. In

a cross-sectional data set a response is measured only once per subject, in contrast to

longitudinal data. Longitudinal (but also clustered or spatial) data require appropriate

statistical tools as it should be possible to model correlation between observations on the

same subject (same cluster or adjacent regions). Since Laird & Ware (1982) mixed models

became a popular and widely used tool to model repeated measurements in the framework

of normal regression models. Before discussing the extension of GLMs to GLMMs, we

recall the main principles of linear mixed models.

Linear Mixed Models (LMMs)

Linear mixed models extend classical linear regression models by incorporating random

effects in the structure for the mean. Assume the data set at hand consists of N subjects.

Let ni denote the number of observations for the ith subject. Y i is the ni × 1 vector of

observations for the ith subject (1 ≤ i ≤ N). The general linear mixed model is given by

Y i = X iβ + Zibi + εi. (4)

β (p × 1) contains the parameters for the p fixed effects in the model; these are fixed,

but unknown, regression parameters, common to all subjects. bi (q × 1) is the vector

with the random effects for the ith subject in the data set. The use of random effects

reflects the belief that there is heterogeneity among subjects for a subset of the regression

coefficients in β. X i (ni× p) and Zi (ni× q) are the design matrices for the p fixed and q

random effects. εi (ni × 1) contains the residual components for subject i. Independence

between subjects is assumed. bi and εi are also assumed to be independent and we follow

the traditional assumption that they are normally distributed with mean vector 0 and

covariance matrices, say D (q× q) and Σi (ni× ni), respectively. Different structures for

these covariance matrices are possible; an overview of some frequently used ones can be

found in Verbeke & Molenberghs (2000). It is easy to see that Y i then has a marginal

normal distribution with mean X iβ and covariance matrix V i = Var(Y i), given by

V i = ZiDZ′i + Σi. (5)

In this interpretation it becomes clear that the fixed effects enter only the mean E[Yij],

whereas the inclusion of subject-specific effects specifies the structure of the covariance

between observations on the same unit.

6

Denote the unknown parameters in the covariance matrix V i with α. Conditional on

α, a closed form expression for the maximum likelihood estimator of β exists, namely

β = (N∑

i=1

X′iV

−1i X i)

−1

N∑i=1

X′iV

−1i Y i. (6)

To predict the random effects the mean of the posterior distribution of the random effects

given the data, bi|Y i, is used. Conditional on α, we have

bi = DZ′iV

−1i (Y i −X iβ), (7)

which can be proven to be the Best Linear Unbiased Predictor (BLUP) of bi (where ‘best’

is again in the sense of minimal mean squared error). For estimation of α maximum likeli-

hood (ML) or restricted maximum likelihood (REML) is used. The expression maximized

by the ML (L1), respectively REML (L2), estimates is given by

L1(α; y1, . . . , yN) = c1 − 1

2

N∑i=1

log |V i| − 1

2

N∑i=1

r′iV

−1i ri (8)

L2(α; y1, . . . , yN) = c2 − 1

2

N∑i=1

log |V i| − 1

2

N∑i=1

log |X ′iV

−1i X i| − 1

2

N∑i=1

r′iV

−1i ri,(9)

where ri = yi − X i

(∑Ni=1 X

′iV iX i

)−1 (∑Ni=1 X

′iV

−1i yi

)and c1, c2 are appropriate

constants. Equations (8) and (9) are maximized using iterative numerical techniques such

as Fisher scoring or Newton-Raphson (check Demidenko, 2004, for full details). In (6) and

(7) the unknown α is then replaced with αML or αREML, leading to the empirical BLUE

for β and the empirical BLUP for bi. For inference regarding the fixed and random effects

and the variance components, appropriate likelihood ratio and Wald tests are explained

in Verbeke & Molenberghs (2000).

3.2 GLMMs: model specification

GLMMs extend GLMs by allowing for random, or subject-specific, effects in the linear

predictor (3). These models are useful when the interest of the analyst lies in the individual

response profiles rather than the marginal mean E[Yij]. The inclusion of random effects

in the linear predictor reflects the idea that there is natural heterogeneity across subjects

in (some of) their regression coefficients. This is the case for the actuarial illustrations

discussed in Section 4. Diggle et al. (2002), McCulloch & Searle (2001), Demidenko (2004)

and Molenberghs & Verbeke (2005) are useful references for full details on GLMMs.

Say we have a data set at hand consisting of N subjects. For each subject i (1 ≤ i ≤ N)

ni observations are available. Given the vector bi with the random effects for subject (or

cluster) i, the repeated measurements Yi1, . . . , Yiniare assumed to be independent with a

7

density from the exponential family

f(yij|bi,β, φ) = exp

(yijθij − ψ(θij)

φ+ c(yij, φ)

), j = 1, . . . , ni. (10)

Similar to (2), the following (conditional) relations hold

µij = E[Yij|bi] = ψ′(θij) and Var[Yij|bi] = φψ

′′(θij) = φV (µij) (11)

where g(µij) = x′ijβ + z

′ijbi. As before, g(.) is called the link and V (.) the variance

function. β (p× 1) denotes the fixed effects parameter vector and bi (q × 1) the random

effects vector. xij (p × 1) and zij (q × 1) contain subject i’s covariate information for

the fixed and random effects, respectively. The specification of the GLMM is completed

by assuming that the random effects, bi (i = 1, . . . , N), are mutually independent and

identically distributed with density function f(bi|α). Hereby α denotes the unknown

parameters in the density. Traditionally, one works under the assumption of (multivariate)

normally distributed random effects with zero mean and covariance matrix determined

by α. Correlation between observations on the same subject arises because they share

the same random effects bi.

The likelihood function for the unknown parameters β, α and φ then becomes (with

y = (y′1, . . . , y

′n)′)

L(β,α, φ; y) =N∏

i=1

f(yi|α, β, φ)

=N∏

i=1

∫ ni∏j=1

f(yij|bi,β, φ)f(bi|α)dbi, (12)

where the integral is with respect to the q dimensional vector bi. When both the data

and the random effects are normally distributed (as in the linear mixed model), the

integral can be worked out analytically and closed-form expressions exist for the maximum

likelihood estimator of β and the BLUP for bi (see (6) and (7)). For general GLMMs,

however, approximations to the likelihood or numerical integration techniques are required

to maximize (12) with respect to the unknown parameters. These will be sketched briefly

in a following subsection and discussed in more detail in Appendix A.

For illustration, we consider a Poisson GLMM with normally distributed random in-

tercept. This GLMM allows explicit calculation of the marginal mean and covariance

matrix. In this way one can clearly see how the inclusion of the random effect models

over-dispersion and within-subject covariance in this example.

Example: a Poisson GLMM Let Yij denote the jth observation for subject i.

Assume that, conditional on bi, Yij follows a Poisson distribution with µij = E[Yij|bi] =

8

exp (x′ijβ + bi) and that bi ∼ N(0, σ2

b ). Straightforward calculations lead to

Var(Yij) = Var(E(Yij|bi)) + E(Var(Yij|bi))

= E(Yij)(exp (x′ijβ)[exp (3σ2

b/2)− exp (σ2b/2)] + 1), (13)

and

Cov(Yij1 , Yij2) = Cov(E(Yij1|bi), E(Yij2|bi)) + E(Cov(Yij1 , Yij2|bi))

= exp (x′ij1

β) exp (x′ij2

β)(exp (2σ2b )− exp (σ2

b )). (14)

Hereby we used the expressions for the mean and variance of a lognormal distribution.

We see that the expression in round parentheses in (13) is always greater than 1. Thus,

although Yij|bi follows a regular Poisson distribution, the marginal distribution of Yij is

over-dispersed. According to (14), due to the random intercept, observations on the same

subject are no longer independent.

GLMMs are appropriate for statistical problems where the modelling and prediction

of individual response profiles is of interest. However, when interest lies only in the pop-

ulation average (and the effect of explanatory variables on it), so-called marginal models

(see Diggle et al., 2002 and Molenberghs & Verbeke, 2005) extend GLMs for indepen-

dent data to models for clustered data. For instance, when using Generalized Estimating

Equations, the effect of explanatory variables on the marginal expectation E[Yij] (instead

of the conditional expectation E[Yij|bi] as in (11)) is specified and –separately– a ‘working’

assumption for the association structure is assumed. The regression parameters in β then

give the effect of the corresponding explanatory variables on the population average. In a

GLMM however these parameters represent the effect of the explanatory variables on the

responses for a specific subject and (in general) they do not have a marginal interpretation.

For, E[Yij] = E[E[Yij|bi]] = E[g−1(x′ijβ + z

′ijbi)] 6= g−1(x

′ijβ).

3.3 Parameter estimation, inference and prediction

In general, the integral in (12) can not be evaluated analytically. The normal-normal case

(normal distribution for the response as well as random effects) is an exception. More

general situations require either model approximations or numerical integration tech-

niques to obtain likelihood-based estimates for the unknown parameters. This paragraph

gives a very brief introduction to restricted pseudo-likelihood ((RE)PL) (Wolfinger &

O’Connell, 1993) and (adaptive) Gauss-Hermite quadrature (Liu & Pierce, 1994) to per-

form the maximum likelihood estimation. Both techniques are available in the commercial

software package SAS and their use will be illustrated later on. The pseudo-likelihood

technique corresponds with the penalized quasi-likelihood (PQL) method of Breslow &

Clayton (1993). Since maximum likelihood techniques are hindered by the integration

over the q-dimensional vector of random effects, a Bayesian implementation of GLMMs

9

is considered as well. Hereby random numbers are drawn from the relevant posterior and

predictive distributions using Markov Chain Monte Carlo (MCMC) techniques. Win-

Bugs allows easy implementation of these models. To make this article self-contained a

first introduction to technical details is bundled in Appendix A. Illustrative code for both

SAS and WinBugs is available on the web 2.

3.3.1 Maximum likelihood approach

(Restricted) Pseudo-likelihood ((RE)PL)

Using a Taylor series the pseudo-likelihood technique approximates the original GLMM

by a linear mixed model for pseudo-data. In this linearized model the maximum likelihood

estimators for the fixed effects and BLUPs for the random effects are obtained using the

well-known theory for linear mixed models (as outlined in Section 3.1). The advantage

of this approach is that a large number of random effects but also crossed and nested

random effects can be handled. A disadvantage is that no true log-likelihood is used.

Therefore likelihood-based statistics should be interpreted with great caution. Moreover

the estimation process is doubly iterative; a linear mixed model is fit, which is an iterative

process, and this procedure is repeated until the difference between subsequent estimates

is sufficiently small. In SAS the macro %Glimmix 3 and procedure Proc Glimmix 4

enable pseudo-likelihood estimation. Justifications of the approach are given by Wolfinger

& O’Connell (1993) and Breslow & Clayton (1993) (via Laplace approximation) where

the approach is called Penalized Quasi-Likelihood (PQL).

(Adaptive) Gauss-Hermite quadrature

Non-adaptive Gauss-Hermite quadrature is an example of a numerical integration tech-

nique that approximates any integral of the form

∫ +∞

−∞h(z) exp (−z2)dz (15)

by a weighted sum, namely

∫ +∞

−∞h(z) exp (−z2)dz ≈

Q∑q=1

wqh(zq). (16)

Here Q denotes the order of the approximation, the zq are the zeros of the Qth order

Hermite polynomial and the wq are corresponding weights. The nodes (or quadrature

2see http://www.econ.kuleuven.be/katrien.antonio3available at http://ftp.sas.com/techsup/download/stat/4experimental version available as an add-on to SAS 9.1

10

points) zq and the weights wq are tabulated in Abramowitz & Stegun (1972, page 924).

By using an adaptive Gauss-Hermite quadrature rule the nodes are rescaled and shifted

such that the integrand is sampled in a suitable range. Details on (adaptive) Gauss-

Hermite quadrature are given in Liu & Pierce (1994) and are summarized in Appendix

A.2. This numerical integration technique still enables for instance a likelihood ratio

test. Moreover the estimation process is just singly iterative. On the other hand, at

present, the procedure can only deal with a small number of random effects which limits

its general applicability. (Adaptive) Gauss-Hermite quadrature is available in SAS via

Proc Nlmixed.

3.3.2 Bayesian approach

The maximum likelihood techniques presented in the previous section are complicated

by the integration over the q-dimensional vector of random effects. Therefore we also

consider a Bayesian implementation of GLMMs, which enables the specification of com-

plicated structures for the linear predictor (combining, for instance, a lot of random effects,

or crossed and nested effects). Another advantage of the Bayesian approach is that various

(other than the normal) distributions can be used for the random effects, whereas in cur-

rent statistical software (like SAS) only normally distributed random effects are available,

though use of the EM algorithm enables other specifications like a Student t-distribution

or a mixture of normal distributions.

The Bayesian approach treats all unknown parameters in the GLMM as random vari-

ables. Prior distributions are assigned to the regression parameters in β and the covariance

matrix D of the normal distribution for the random effects. Since the posterior distribu-

tions involved in GLMMs are typically numerically and analytically intractable, posterior

and predictive inference is based on drawing random samples from the relevant posterior

and predictive distributions with Markov Chain Monte Carlo (MCMC) techniques. An

early reference to Gibbs sampling in GLMMs is Zeger & Karim (1991).

For the examples in Section 4, the following distributional and prior specifications are

used (canonical link for ease of notation):

[Y |b] = exp {y′(Xβ + Zb)− 1′ψ(Xβ + Zb) + 1

′c(y)}, (17)

where Y = (Y′1, . . . , Y

′N)

′, b = (b

′1, . . . , b

′N)

′and X, Z are the corresponding design

matrices. Further,

[b] ∼ N(0, blockdiag(G)), [β] ∼ N(0,F ), [G] ∼ Inv-Wishartν(B), (18)

where F is a diagonal matrix with large entries and B has the same dimensions as G.

These specifications result in the following full conditionals

[β|.] ∝ exp {y′Xβ − 1′ψ(Xβ + Zb)− 1

2β′F−1β}, (19)

11

[b|.] ∝ exp {y′Zb− 1′ψ(Xβ + Zb)− 1

2b′G−1b}, (20)

[G|.] ∝ Inv-Wishartν+N(B +N∑

i=1

bib′i). (21)

For a single variance component, an inverse gamma prior is used, which results in an

inverse gamma full conditional. In the sequel of this article, the WinBugs software is

used for Gibbs sampling. Zhao et al. (2005) give more technical details on Gibbs sampling

in GLMMs.

3.3.3 Inference and Prediction

In actuarial regression problems (see Section 2) accurate prediction for individual risk

classes is of major concern. Since Bayesian statistics allows simulation from the full

predictive distribution of any quantity of interest, this approach is the most appealing

from this point of view. In Section 4.2.2 also predictions obtained with Proc Glimmix

and Proc Nlmixed are displayed. With Proc Glimmix we report g−1(ηi,ni+1) for

the predicted value and√

Var(g−1(ηi,ni+1 − z′i,ni+1bi)) for the prediction error. Hereby

ηi,ni+1 = x′i,ni+1β +z

′i,ni+1bi and the delta method is applied to Var

[β

bi − bi

], obtained

from the final Proc Mixed call. With Proc Nlmixed, predicted values are computed

from the parameter estimates for the fixed effects and empirical Bayes predictors for

the random effects. Standard errors of prediction for some function of the fixed and

random effects (say, g−1(x′i,ni+1β + z

′i,ni+1bi)) are again obtained with the delta method

and an approximate prediction variance matrix for

[β

bi

]. For Var(β) the inverse of an

approximation of the corresponding Fisher information matrix is used. For the variance of

bi, Proc Nlmixed uses an approximation to Booth & Hobert’s (1998) conditional mean

squared error of prediction. The predictions and prediction errors reported in Section

4.2.2, obtained with PQL and G-H respectively, are very close.

4 Applications

4.1 A GLMM interpretation of traditional credibility models

The credibility ratemaking problem is concerned with the determination of a risk premium

for a group of contracts which combines the claims experience related to that specific

group and the experience regarding related risk classes. This approach is justified since

the risks in a certain class have characteristics in common with risks in other classes,

but they also have their own risk class specific features. For a general introduction to

12

classical credibility theory, we refer to Dannenburg et al. (1996). Using linear mixed

models Frees et al. (1999) already gave a longitudinal data analysis interpretation of

the well-known credibility models of Buhlmann (1967,1969), Buhlmann-Straub (1970),

Hachemeister (1975) and Jewell (1975). They explained how to specify the fixed (β) and

random effects (bi) for every subject or risk class i (i = 1, . . . , N) and used β and bi (as

in (6) and (7)) to derive the Best Linear Unbiased Predictor for the conditional mean of a

future observation (E[Yi,ni+1|bi]). For the above mentioned credibility models, this BLUP

corresponds with the classical credibility formulas (as in Dannenburg et al., 1996).

However, the normal-normal model (normality for both responses and random effects)

will not always be plausible for the data at hand (which can be, for instance, counts,

binary or skewed data). Therefore it is useful to revisit the credibility models in the

context of GLMs and to consider their specification as a GLMM. In this way estimators

and predictors will be used that take the distributional features of the data into account.

For GLMMs (in general) no closed-form expressions for β and bi exist and one has to rely

on the techniques presented in Section 3.3 to derive a predictor for E[Yi,ni+1|bi].

The interpretation of actuarial credibility models as (generalized) linear mixed mod-

els has several advantages over the traditional approach. For instance, the inferential

tools from the theory of mixed models can be used to check the models. Moreover, the

likelihood-based estimation of fixed effects and unknown parameters in the covariance

structure is well-developed in the framework of (G)LMMs, which provides an alternative

for the traditional moment-based estimators used in actuarial literature (see Dannenburg

et al., 1996).

Whereas this paper puts focus on statistical aspects of GLMMs and their use in the

modelling of real data sets, Purcaru & Denuit (2003) studied the kind of dependence

that arises from the inclusion of random effects in credibility models for frequency counts,

based on GLMs.

4.1.1 Mixed model specification of well-known credibility models

Interpreting traditional credibility models in the context of GLMMs implies that the

additive regression structure in terms of fixed and subject-specific (or risk class specific)

effects is specified on the scale of the linear predictor, namely

g(µij) = ηij = x′ijβ + z

′ijbi. (22)

Hereby i (i = 1, . . . , N) denotes the subject, for instance a policy(holder) or risk class,

and j refers to its jth measurement, unless it is stated otherwise. The link function g(.)

and variance function V (.) are determined by the chosen GLM. Next to the above men-

tioned models of Buhlmann (1967, 1969), Buhlmann-Straub (1970), Hachemeister (1975)

and Jewell (1975), we also consider the crossed classification model of Dannenburg et al.

(1996), which is illustrated on a real data set in Section 4.2.1.

13

1. Buhlmann’s model

Expression (22) becomes g(µij) = ηij = β + bi (for i = 1, . . . , N). Thus, on the scale of

the linear predictor, a single fixed effect β denotes the overall mean and the random effect

bi denotes the deviation from this mean, specific for contract or policyholder i.

2. Buhlmann-Straub’s model

Buhlmann’s model is extended by introducing (known) positive weights. In the GLMM

specification weights are introduced by replacing φ with φ/wij in f(yij|β, bi, φ) (see (10)).

The structure of the linear predictor from Buhlmann’s model remains unchanged. The

importance of including weights is already discussed in Kaas et al. (1997).

3. Hachemeister’s model

Hachemeister’s model is a credibility regression model and generalizes Buhlmann’s model

by introducing collateral data. The structure for the linear predictor is obtained by tak-

ing x′ij = z

′ij in (22). For instance, xit = zit = (1 t)

′leads to the linear trend model

considered in Frees et al. (1999).

4. Jewell’s hierarchical model

This hierarchical model classifies the original set of N policies or subjects according to a

certain number of risk factors. Consider for instance the mixed model formulation for a

two-stage hierarchy. We have that g(µijt) = ηijt = β + bi + bij. Hereby i denotes the risk

factor at the highest level and j denotes the categories of the second risk factor. t indexes

the repeated measurements for this (i, j) combination. bi is a random effect for the first

risk factor. The random effect bij characterizes category j of the second risk factor within

the ith category of the highest risk factor. More than two nestings can be dealt with in

the same manner.

5. Dannenburg’s crossed classification model

Crossed classification models are an extension of the hierarchical models studied by Jewell

(1975) and also allow risk factors to occur next to each other, instead of fully nested. For

instance, in the two-stage hierarchy described above, there is no risk factor which char-

acterizes the second stage, without interacting with the first stage. Dannenburg’s model

(Dannenburg et al., 1996) allows a combination of crossed and nested random effects and

specifies the linear predictor for the two-way model as ηijt = β + b(1)i + b

(2)j + b

(12)ij , where

(1) and (2) refer to the two different risk factors involved.

14

4.2 Data Examples

Three examples are discussed where GLMMs are used to model real data. In a first

example Dannenburg’s crossed classification model is considered. Afterwards, the data

introduced in Section 2 are analyzed.

4.2.1 Dannenburg’s Crossed Classification Model

We consider the data from Dannenburg et al. (1996, page 108). The crossed classification

model is illustrated on a two-way data set made available by a credit insurer. The data are

payments of this credit insurer to several banks in order to cover losses caused by clients

who were not able to pay off their private loans. The observations have been categorized

according to marital status of the debtor and the time he worked with his current employer.

On every cell (i.e. a status-experience combination) repeated measurements are available.

Exploratory plots are given in Figure 5. Scollnik (2000) considered the same data set to

illustrate a Bayesian implementation of the crossed classification model (under normality

for the response). We believe that it is illustrative to analyze these data again, since

we explicitly consider this credibility model as a GLMM and illustrate how to estimate

parameters and risk premia using likelihood-based as well as Bayesian techniques.

Let i denote the marital status of the debtor (1=single, 2=divorced and 3=other) and

j the time he worked with his current employer (1=less than 2 years, 2=between 2 and

10 years and 3=more than 10 years). Dannenburg et al. (1996) modelled these data as

Yijt = β + b(1)i + b

(2)j + b

(12)ij + εijt, where t refers to the tth observation in cell (i, j). The

original crossed classification model assumes all the random components to be mutually

independent with zero mean and δ1 = Var(b(1)i ), δ2 = Var(b

(2)j ), δ12 = Var(b

(12)ij ) and

σ2 = Var(εijt). Dannenburg et al. (1996) had to rely on typical (complicated) credibility

calculations to obtain estimates for the risk premia in terms of credibility factors and

weighted averages of the observations in the different cells or levels. These are functions

of the unknown variance components and overall mean (β) for which moment-based es-

timators are proposed. The theory of mixed models is useful here since it provides a

unified framework to perform the estimation of the regression parameter β and variance

components δ1, δ2, δ12 and σ2, together with the prediction of the random effects.

As illustrated clearly by Figure 5, the data are right-skewed. In contrast to Scollnik

(2000) we analyzed the data with a gamma GLMM with logarithmic link function and

normality for the random effects. We initially specified the linear predictor as ηijt =

β + b(1)i + b

(2)j + b

(12)ij , but the variance component corresponding with the interaction term

was put equal to zero by the procedure. This is in line with the findings in Dannenburg et

al. (1996, page 106). Thus, the model is reduced to a two-way model without interaction

and a linear predictor of the form ηijt = β + b(1)i + b

(2)j . We fit this model using the

penalized quasi-likelihood technique described in Section 3.3.1 and its implementation

15

via the SAS macro %Glimmix and the SAS procedure Proc Glimmix. The results

are given in Table 1 and 2. Note that a test for the need of the random effects involves

a boundary problem (H0 : δ1 = 0 versus H1 : δ1 > 0, for instance) and would, with

our model specification, rely on Monte Carlo simulations. For prediction purposes, a

Bayesian implementation of the model is extremely useful. Figure 6 shows histograms for

simulations from the predictive distribution of Yij,nij+1 for every (i, j) combination.

0 200 400 600 800

020

4060

Dannenburg’s Cross-Classification Model

Payment

020

040

060

080

0

1 2 3 4 5 6 7 8 9

Dannenburg’s Cross-Classification Model

Combination of Experience and Status

Pay

men

t

Figure 5: Histogram of ‘Payment’ (whole database) and boxplots of ‘Payment’ for each

(i, j) combination (1=(1,1), 2=(1,2), 3=(1,3), 4=(2,1), 5=(2,2), 6=(2,3), 7=(3,1),

8=(3,2), 9=(3,3)). i gives the marital status (1=single, 2=divorced, 3=other) and j

the time worked with current employer (1 : < 2 yrs, 2 : ≥ 2 and ≤ 10 yrs, 3 : > 10 yrs).

Effect Parameter Estimate (s.e.)

Fixed-effects parameter

Intercept β 5.475 (0.146)

Variance parameters

Class i δ1 0.014 (0.018)

Class j δ2 0.046 (0.045)

Table 1: Gamma-normal mixed model with logarithmic link function: estimates for fixed

effect and variance components (under REML), credit insurer data. Results obtained with

Proc Glimmix in SAS.

4.2.2 Workers’ Compensation Insurance: Frequencies

Background for this data set is given in Section 2.1 and exploratory plots are shown in

Figure 1 and 2. We analyze the data using mixed models and build up suitable regression

models in this context. Since the data are frequency counts and interest lies in predictions

16

j = 1 j = 2 j = 3

i = 1 183 239 275

i = 2 176 230 265

i = 3 215 281 323

Table 2: Gamma-normal mixed model: estimated risk premia based on the estimate for

the fixed effect and predictions for the random effects using SAS Proc Glimmix, credit

insurer data.

0 200 400 600 800 1200

050

015

00

Premium 1

0 500 1000 1500

010

0025

00

Premium 2

0 500 1000 1500

010

0020

00

Premium 3

0 200 400 600 800 1000

050

015

00

Premium 4

0 200 400 600 800 1000

050

015

00

Premium 5

0 500 1000 1500 20000

1000

2500

Premium 6

0 200 400 600 800 1200

050

015

00

Premium 7

0 500 1000 1500

010

0020

00

Premium 8

0 500 1000 1500 2000

050

015

00

Premium 9

Figure 6: Predictive distributions for different risk classes, credit insurer data. Burn-in

of 50,000 simulations, followed by another 50,000 simulations.

regarding individual risk classes, our analysis uses a Poisson GLMM. Heterogeneity across

risk classes and correlation between observations on the same class further motivate the

inclusion of random effects in the linear predictor of the Poisson GLM.

The following models are considered

Yij|bi ∼ Poisson(µij)

where log (µij) = log (Payrollij) + β0 + β1Yearij + bi,0 (23)

versus log (µij) = log (Payrollij) + β0 + β1Yearij + bi,0 + bi,1Yearij. (24)

Hereby Yij represents the jth measurement on the ith subject of the response Count. β0

and β1 are fixed effects and bi,0, versus bi,1, is a risk class specific intercept, versus slope.

It is assumed that bi = (bi,0, bi,1)′ ∼ N(0, D) and that, across subjects, random effects

17

are independent. The results of both a maximum likelihood and a Bayesian analysis are

given in Table 3. The models were fitted to the data set without the observed Counti7,

to enable out-of-sample prediction later on. In Table 3, δ0 = Var(bi,0), δ1 = Var(bi,1) and

δ0,1 = δ1,0 = Cov(bi,0, bi,1). Proc Glimmix did not converge with model specification

(24), together with a general structure for the covariance matrix D of the random effects.

For the adaptive Gauss-Hermite quadrature 20 quadrature points were used in SAS Proc

Nlmixed, though for example, an analysis with 5, 10 and 30 quadrature points gave

practically indistinguishable results. For the prior distributions in the Bayesian analysis

we used the specifications from Section 3.3.2. Four parallel chains were run and for each

chain a burn-in of 20,000 simulations was followed by another 50,000 simulations. As

reported by many authors (see for instance Zhao et al., 2005) centering of covariates and

hierarchical centering greatly improves mixing of the chains and speed of simulation.

PQL adaptive G-H Bayesian

Est. SE Est. SE Mean 90% Cred. Int.

Model (23)β0 -3.529 0.083 -3.557 0.084 -3.565 (-3.704, -3.428)β1 0.01 0.005 0.01 0.005 0.01 (0.001, 0.018)δ0 0.790 0.110 0.807 0.114 0.825 (0.648, 1.034)

Model (24)β0 -3.532 0.083 -3.565 0.084 -3.585 (-3.726, -3.445)β1 0.009 0.011 0.009 0.011 0.008 (-0.02, 0.04)δ0 0.790 0.111 0.810 0.115 0.834 (0.658, 1.047)δ1 0.006 0.002 0.006 0.002 0.024 (0.018, 0.032)δ0,1 / / 0.001 0.01 0.006 (-0.021, 0.034)

Table 3: Workers’ compensation data (frequencies): results of maximum likelihood and

Bayesian analysis. REML is used in PQL.

To check whether model (24) can be reduced to (23), a likelihood ratio test for the

need of random slopes was used and resulted in an observed value of 74=4394-4320 (with

adaptive G-H quadrature, 20 quadrature points). Figure 7 illustrates that the fitted values

obtained with the preferred model (24) are close to the real observed values. To illustrate

prediction with this model, Table 4 compares the predictions for some selected risk classes

with the observed values. These risk classes are the same as in Klugman (1992) and

Scollnik (1996). The predicted values are calculated as exp (µi7) (i = 11, 20, 70, 89, 112).

The reported prediction errors are obtained as described in Section 3.3.3. As mentioned

in the same section, when focus lies on prediction, the Bayesian approach is the most

useful, since it provides the full predictive distribution of Yi7. For the selected risk classes,

18

Workers’ Compensation Insurance: Frequencies

ObservedF

itted

0 50 100 150 200

050

100

150

200

Figure 7: Workers’ compensation data (frequencies): scatterplot of observed counts versus

fitted values for Model (24), via SAS Proc Nlmixed.

these predictive distributions are illustrated in Figure 8.

Actual Values Expected number of claimsPQL adaptive G-H Bayesian

Class Payrolli7 Counti7 Mean s.e. Mean s.e. Mean 90% Cred. Int.

11 230 8 11.294 2.726 11.296 2.728 12.18 (5,21)20 1,315 22 33.386 4.109 33.396 4.121 32.63 (22,45)70 54.81 0 0.373 0.23 0.361 0.23 0.416 (0,2)89 79.63 40 47.558 5.903 47.628 6.023 50.18 (35,67)112 18,810 45 33.278 4.842 33.191 4.931 32.66 (21,46)

Table 4: Workers’ compensation data (frequencies): predictions for selected risk classes.

4.2.3 Workers’ Compensation Insurance: Losses

We consider the data from Section 2.2. Frees et al. (2001) focused on the longitudinal

character of the data and modelled the logarithmic transformation of PP=Loss/Payroll

using linear mixed models. Our analysis as well takes the longitudinal character of the

data into account and considers inference and prediction regarding individual risk classes.

Use is made, however, of a gamma GLMM; in this way no transformation of the data is

required. Loss is the response variable and Payroll is used as an offset. Figure 9 (left)

shows a gamma QQplot for Loss, without taking into account the regression structure of

19

0 10 20 30 40

020

0040

0060

0080

00

Num, Class 11

10 20 30 40 50 60 70

010

0030

0050

00

Num, Class 20

0 2 4 6 8 10

010

000

2000

030

000

Num, Class 70

20 40 60 80 100

020

0040

0060

0080

00

Num, Class 89

10 20 30 40 50 60 70

010

0020

0030

0040

0050

00

Num, Class 112

Figure 8: Workers Compensation Insurance (Counts): predictive distributions for selec-

tion of risk classes.

the data. The following models are considered

Yij|bi ∼ Γ(ν, µij/ν)

where log (µij) = log (Payrollij) + β0 + bi,0 (25)

versus log (µij) = log (Payrollij) + β0 + β1Yearij + bi,0 (26)

and log (µij) = log (Payrollij) + β0 + β1Yearij + bi,0 + bi,1Yearij. (27)

The gamma density function is specified as f(y) = 1Γ(ν)

(νyµ

)ν

exp(−νy

µ

)1y. The speci-

fication in (27) did not lead to convergence of the SAS procedures. Structure (26) is

the preferred choice for the linear predictor. Table 5 contains the results of a maximum-

likelihood and Bayesian analysis. For the parameter ν a uniform prior was specified, other

prior specifications were as in Section 3.3.2. Fitted values against real observations are

plotted in Figure 9 (right).

For this example the gamma GLMM has an important advantage over the normal

mixed model. A linear mixed model is only applicable after a log-transformation of the

data (as in the analysis of Frees et al., 2001). However, a correct back-transformation of

predictions on the logarithmic scale to the original scale is not straightforward. When

using a gamma GLMM this is not an issue, since the data are used in their original form.

20

Quantiles Gamma (MOM)

Loss

0 5*10^6 10^7 1.5*10^7 2*10^7 2.5*10^7

05*

10^6

10^7

1.5*

10^7

2*10

^7

Workers’ Compensation Insurance: Losses

Observed

Fitt

ed

0 5*10^6 10^7 1.5*10^7 2*10^7

05*

10^6

10^7

1.5*

10^7

2*10

^7

Figure 9: Workers’ compensation data (Losses): gamma QQplot of loss.

PQL adaptive G-H Bayesian

Est. SE Est. SE Mean 90% Cred. Int.

β0 -4.172 0.091 -4.148 0.091 -4.147 (-4.298, -3.996)β1 0.042 0.012 0.042 0.012 0.042 (0.022, 0.062)δ1 0.915 0.128 0.912 0.127 0.938 (0.741, 1.174)

Table 5: Workers’ compensation data (Losses): results of maximum likelihood and

Bayesian analysis. REML is used in PQL.

4.3 Additional comments

GLMMs in Claims Reserving

In Antonio et al. (2006) a mixed model for loss reserving is presented which is based

on a data set containing individual claim records. In this way the authors develop an

alternative for claims reserving models based on traditional run-off triangles, which are

only a summary of an underlying data set containing the individual figures. A general

mixed model is then applied to the logarithmic transformed data. Generalized linear

mixed models are a valuable alternative, since they do not require a transformation of the

data and allow to model the data using (other) distributions from the exponential family.

GLMMs in Credit Risk Modelling

McNeil & Wendin (2005) provide another interesting –financial– application of GLMMs.

GLMMs with dynamic random effects are used for the modelling of portfolio credit default

21

risk. Details on a Bayesian implementation of the models are included.

5 Summary

This paper discusses generalized linear mixed models as a tool to model actuarial longi-

tudinal and (other forms of) clustered data. In Section 3 the most important concepts

on model formulation, estimation, inference and prediction are summarized. In this way

it is our intention to make this literature better accessible to actuarial practitioners and

researchers. Both a maximum likelihood and Bayesian approach are presented. Section 4

describes various applications of GLMMs in the domains of credibility, non-life ratemak-

ing, credit risk modelling and loss reserving.

A Maximum likelihood approach: technical details

A.1 (Restricted) Pseudo-Likelihood ((RE)PL)

Consider the response vector Y i (ni × 1) for subject or cluster i (i = 1, . . . , N). Let β

and bi be known estimates of β (p × 1) and bi (q × 1). Assume that the random effects

have mean 0 and covariance matrix D, and that the conditional covariance matrix for Y i

is specified as

Var[Y i|bi] = A1/2µi

RiA1/2µi

. (28)

Hereby Aµi(ni × ni) is a diagonal matrix with V (µij) (j = 1, . . . , ni) on the diagonal

(see (11)) and Ri (ni × ni) has a structure to be specified by the analyst (for instance

Ri = φIni×ni). Define ei = Y i−µi and consider a first order Taylor series approximation

of ei about the known estimates β and bi. We then get

ei = Y i − µi − ∆i(ηi − ηi). (29)

Hereby g(µi) = ηi = X iβ + Zibi and ∆i (ni × ni) is a diagonal matrix with the first

order derivative of g−1(.) evaluated in the components of ηi as diagonal elements.

Wolfinger & O’Connell (1993) then approximate the distribution of ei given β and bi

by a normal distribution with the same first two moments as the conditional distribution

of ei|β, bi. Substituting µi for µi in Aµi, this approximation leads to

νi|β, bi ∼ N(X iβ + Zibi, GiA

1/2µi

RiA1/2µi

Gi

)

for νi = g(µi) + Gi(Y i − µi). (30)

Hereby Gi is a diagonal matrix with g′(µi) on the diagonal. As such, the pseudo data

vector νi (which is a Taylor series approximation of g(Y i) around µi) follows a linear

22

mixed model as specified in (4) with bi ∼ N(0,D) and εi ∼ N(0, GiA1/2µi

RiA1/2µi

Gi).

Following the approach from linear mixed models, the unknown parameters in D and

Ri are estimated by maximizing the log-likelihood (L1) or restricted log-likelihood (L2)

function, with respect to the pseudo-data. Once estimates for the unknown parameters

in D and Ri are obtained, the (empirical) maximum likelihood estimator for β and the

Best Linear Unbiased Predictor for bi are obtained with the well-known theory from

linear mixed models. The (restricted) pseudo-likelihood procedure is repeated until the

difference between subsequent estimates for D and Ri is sufficiently small. As such,

parameter estimates are computed by iteratively applying standard theory from linear

mixed models.

A.2 (Adaptive) Gauss-Hermite quadrature

Full details and references on Gauss-Hermite quadrature for mixed models are in Liu &

Pierce (1994). Consider first the situation of a single random effect bi. The contribution

of subject i to the marginalized likelihood is given by

∫ ni∏j=1

f(yij|bi, β, φ)f(bi|σ)dbi, (31)

where the random effect bi is assumed to follow a normal distribution with zero mean and

standard deviation σ. After a reparameterization to δi = σ−1bi, (31) can be rewritten as

∫ ni∏j=1

f(yij|σδi,β, φ)φ(δi; 0, 1)dδi (32)

where φ(δi; 0, 1) denotes the standard normal density and f(yij|σδi, β, φ) follows the same

GLM as before but now with g(µij) = x′ijβ + zijσδi.

Non-adaptive Gauss-Hermite quadrature approximates an integral of the form∫ +∞−∞ h(z) exp (−z2)dz by a weighted sum, namely

∫ +∞

−∞h(z) exp (−z2)dz ≈

Q∑q=1

wqh(zq). (33)

Q denotes the order of the approximation, the zq are the zeros of the Qth order Hermite

polynomial and the wq are corresponding weights. The nodes (or quadrature points) zq

and the weights wq are tabulated in Abramowitz & Stegun (1972, page 924).

The quadrature points used in (33) do not depend on h. As such, it is possible that only

very few nodes lie in the region where most of the mass of h is, which would lead to poor

approximations. Using an adaptive Gauss-Hermite quadrature rule the nodes are rescaled

and shifted such that the integrand is sampled in a suitable range. Assume h(z)φ(z; 0, 1)

23

to be unimodal and consider the numerical integration of∫ +∞−∞ h(z)φ(z; 0, 1)dz. Let µ and

σ be

µ = mode [h(z)φ(z; 0, 1)] and σ2 =

[− ∂2

∂z2ln (h(z)φ(z; 0, 1))

∣∣∣z=µ

]−1

. (34)

Acting as if h(z)φ(z; 0, 1) were a Gaussian density, µ and σ would be the mean and variance

of this density. The quadrature points in the adaptive procedure, z?q , are centered at µ

with spread determined by σ, namely

z?q = µ +

√2σzq (35)

with (q = 1, . . . , Q). Now rewrite∫ +∞−∞ h(z)φ(z; 0, 1)dz as

∫ +∞−∞

h(z)φ(z;0,1)φ(z;µ,σ)

φ(z; µ, σ)dz,

where φ(z; µ, σ) is the Gaussian density function with mean µ and variance σ2. Using

simple manipulations it is easy to see that for a suitably regular function v

∫ +∞

−∞v(z)φ(z; µ, σ)dz =

∫ +∞

−∞v(z)(2πσ2)−1/2 exp

(−1

2

(z − µ

σ

)2)

dz

=

∫ +∞

−∞

v(µ +√

2σz)√π

exp(−z2

)dz, (36)

which can be approximated using (33). Applying a similar quadrature formula to∫ +∞−∞ h(z)φ(z; 0, 1)dz and replacing µ and σ by µ and σ, we get

∫ +∞

−∞h(z)φ(z; 0, 1)dz ≈

√2σ

Q∑q=1

wq exp (z2q )φ(z?

q ; 0, 1)h(z?q ) =

Q∑q=1

w?qh(z?

q ), (37)

with the adaptive weights w?q :=

√2σwq exp (z2

q )φ(z?q ; 0, 1). (37) is called an adaptive

Gauss-Hermite quadrature formula and can be used to approximate (32).

When bi is a q dimensional vector of random effects, a cartesian product rule based

on (33) (non-adaptive) or (37) (adaptive) can be used to approximate (15), leading to

L(β,α, φ; y) ≈N∏

i=1

Q∑i1=1

wi1

Q∑i2=1

wi2 . . .

Q∑iq=1

wiq

ni∏j=1

f(yij|zi1i2...iq ,β, φ)

, (38)

where Q is the order of the numerical quadrature. In this expression the (adaptive)

quadrature weights wik (k = 1, . . . , q) and nodes zi1i2...iq (a vector with elements (zi1 , . . . , ziq))

depend on the unknown parameters in β, α and φ. Numerical techniques (e.g. Newton-

Raphson) lead to α, β and φ by maximizing (38) over the unknown parameters. Ap-

proximate standard errors for β and α are obtained from the inverse Hessian, evaluated

at the estimates.

As with general linear mixed models, it is most natural to use Bayesian methods to

obtain predictors for the random or subject-specific effects bi (i = 1, . . . , N). Under

24

linear mixed models the posterior distribution bi|Y i is multivariate normally distributed,

which allows a closed-form expression for the posterior mean. For general linear mixed

models this is no longer true and therefore it is customary to use the posterior mode

(rather than the mean) to predict bi. When using adaptive Gauss-Hermite quadrature no

extra calculations are required since the mode of f(yi|bi)f(bi), which equals the mode of

f(bi|yi), is computed during the numerical integration.

25

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